Question

question
find the perimeter of \\( \triangle mno \\). round your answer to the nearest tenth if necessary. figures are not
necessarily drawn to scale.

Explanation:

Step1: Determine similarity of triangles

Since the angles of \(\triangle LJK\) and \(\triangle MNO\) are equal (\(51^{\circ},61^{\circ},68^{\circ}\)), the two triangles are similar by the AA (Angle - Angle) similarity criterion.

Step2: Find the scale factor

The side of length \(9\) in \(\triangle LJK\) corresponds to the side of length \(23.1\) in \(\triangle MNO\). The scale factor \(k=\frac{23.1}{9}=2.566\cdots\)

Step3: Find the lengths of the other sides of \(\triangle MNO\)

The side of length \(11\) in \(\triangle LJK\) corresponds to the side of length \(21\) in \(\triangle MNO\) (already given). The side of length \(10\) in \(\triangle LJK\) corresponds to side \(x\) in \(\triangle MNO\). Using the scale factor, \(x = 10\times\frac{23.1}{9}=\frac{231}{9}=25.7\)

Step4: Calculate the perimeter of \(\triangle MNO\)

Perimeter \(P=21 + 23.1+25.7\)
\(P=69.8\)

Answer:

\(69.8\)